Does anyone know anything about fractals and specifically, the Haussdorff dimension and measure?

I'm going to write a small paper about this and I know the definition of the Haussdorff measure, but I would like some examples to learn from.

The Haussdorff dimension is a dimension defined so that the dimension of a set doesn't have to be an integer, for example the dimension of fractals often is of the form log(a)/log(b).

Well, use some fractals that have been published:

Sierpinski Triangle: dimension is (log 3) / (log 2)

Sierpinski Carpet: dimension is (log 8) / (log 2)

Look for others (Mandelbrot set, Julia set, etc.) There's a wonderful World Wide Web with a very large set of references for you to find (but the Hausdorff dimension of this, or any other, finite set is, alas, an integer).

Regards,

Dave

03-15-2005

Zach L.

I've played around with fractals (and chaotic systems) a little bit, but not much to be of any help. At any rate, the place I'd look for information on this sort is arXiv.org.

What I want to do is to derive the log(N)/log(r) formula from the definition.

Quote:

arXiv.org

Seems like a really nice page. Thanks! Most of the stuff is way above my level, though.

03-16-2005

Dave Evans

Quote:

Originally Posted by Sang-drax

What I want to do is to derive the log(N)/log(r) formula from the definition.

Seems like a really nice page. Thanks! Most of the stuff is way above my level, though.

To me, one of the most interesting things about chaos theory (including characterization of fractals) is that fairly simple mathematical formulas can give rise to incredibly complicated structures.

The beauty of Mandelbrot and Julia sets can be appreciated by most of us without understanding the mathematical principles. Getting into it a little deeper is not worth the effort to some people, but if you really like that kind of stuff (not just the pretty pictures), it can be really exciting and rewarding. There is some math.

The classic book "Chaos: Making a New Science", by James Gleick, is pretty digestable, even without advanced math. Lots of excerpts and examples from that book and others can be found on the Web.